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Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars

Christopher-Lloyd Simon

Abstract

A filoop is a generic immersion of a circle in a closed oriented surface, whose complement is a disjoint union of discs, considered up to orientation preserving diffeomorphisms. It gives rise to a chord diagram C which has an interlace graph G, called a chordiagraph. For a graph G with even degrees, we compute a quantity mg(G) which yields, for every chord diagram $C$ with interlace graph G, the minimal genus of filoops with chord diagram C. If mg(G)=0 then C admits exactly two framings of genus 0, corresponding to spheriloops. After recalling the Cunningham factorisation of connected graphs, we describe a canonical factorisation of filoops into spheric sums followed by toric sums, for which the genus is additive. This is analogous to the factorisation of compact connected 3-manifolds along spheres and tori. We describe unambiguous context-sensitive grammars generating the set of all graphs and with mg(G)=0 and deduce stability properties with respect to spheric and toric factorisations. Similar results hold for chordiagraphs with mg(G) = 0 and their corresponding spheriloops.

Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars

Abstract

A filoop is a generic immersion of a circle in a closed oriented surface, whose complement is a disjoint union of discs, considered up to orientation preserving diffeomorphisms. It gives rise to a chord diagram C which has an interlace graph G, called a chordiagraph. For a graph G with even degrees, we compute a quantity mg(G) which yields, for every chord diagram with interlace graph G, the minimal genus of filoops with chord diagram C. If mg(G)=0 then C admits exactly two framings of genus 0, corresponding to spheriloops. After recalling the Cunningham factorisation of connected graphs, we describe a canonical factorisation of filoops into spheric sums followed by toric sums, for which the genus is additive. This is analogous to the factorisation of compact connected 3-manifolds along spheres and tori. We describe unambiguous context-sensitive grammars generating the set of all graphs and with mg(G)=0 and deduce stability properties with respect to spheric and toric factorisations. Similar results hold for chordiagraphs with mg(G) = 0 and their corresponding spheriloops.
Paper Structure (25 sections, 38 theorems, 18 equations, 25 figures)

This paper contains 25 sections, 38 theorems, 18 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Homological invariants of filoops and their interlace graphs
  3. Unique factorisations of graphs, chord diagrams and filoops
  4. Generating grammars for Gaussian graphs and spheriloops
  5. Section 1.
  6. Section 2.
  7. Section 3.
  8. Enumeration.
  9. Polynomial invariants.
  10. Spectral properties.
  11. Factorisations of $3$-manifolds
  12. Knots and singularities.
  13. Homological invariants of filoops and their interlace graphs
  14. Filoops correspond to framed chord diagrams
  15. Intersection form of the ribbon-graph
  16. ...and 10 more sections

Key Result

Corollary 2

Fix a chordiagraph $G$ satisfying EveN1 with Rosenstiehl form $\mathop{\mathrm{\mathfrak{r}}}\nolimits$. For every chord diagram $C$ with interlace graph $G$, the minimal genus of its framings equals the minimum of the half-rank over the space of bicolour forms $c_\chi\colon V_G\times V_G\to \{0,1\} Fix a chordiagraph $G$ satisfying EveN1 and EveN2 with Rosenstiehl cocycle $\mathop{\mathrm{\mathfr

Figures (25)

  • Figure 1: A spheriloop, its corresponding framed chord diagram, and its interlace graph.
  • Figure 2: A connected (Gaussian chordia)graph and its reduced GLT factorisation.
  • Figure 3: Based filoops $\alpha$ and $\beta$, their spheric sum $\alpha \otimes \beta$ and plumbing $\alpha \asymp \beta$.
  • Figure 4: The unique framing of genus $0$ and its corresponding spheriloop.
  • Figure 5: The filoops with framed chord diagrams $a_\infty a_0$ and $a_\infty b_0 b_\infty a_0$ and $a_\infty b_0 c_\infty a_0 b_\infty c_0$ and $b_0 a_\infty b_\infty a_0$ and $a_0 b_0 b_\infty c_\infty c_0 a_\infty d_0 d_\infty e_\infty f_\infty f_0 e_0$ and $a_\infty b_0 c_\infty d_\infty e_0 f_\infty b_\infty a_0 f_0 c_0 d_0 e_\infty$.
  • ...and 20 more figures

Theorems & Definitions (98)

  • Definition 1: Rosentiehl form & Gaussian graphs
  • Corollary 2: Minimal genus
  • Corollary 3: Spheriloops with prescribed chord diagram
  • Definition 4
  • Definition 5
  • Definition 6: Sketch
  • Corollary 7: Spheric-sum factorisation
  • Theorem 8: Essential-plumbing factorisation
  • Corollary 9
  • Theorem 10: Grammatical generation of Gaussian GLT
  • ...and 88 more